A370776 The maximum number of alive cells reached in Conway's Game of Life when starting with the first n primes in Ulam's spiral; or -1 if no such maximum exists.

1, 2, 3, 4, 5, 6, 7, 65, 56, 120, 56, 28, 133, 30, 160, 46, 24, 24, 25, 28, 30, 31, 31, 32, 32, 32, 35, 74, 39, 38, 38, 38, 39, 40, 42, 319, 319, 319, 319, 319, 46, 129, 93, 50, 50, 72, 72, 72, 72, 72, 72, 53, 53, 56, 56, 851, 851, 167, 167, 167, 167, 391

Offset

1,2

Comments

The initial alive cells are at coordinates x=A214664(i), y=A214665(i) for i=1..n.

For the first 7 terms of this sequence we have a(n)=n since those initial configurations do not lead to complex enough patterns that increase the number of alive cells beyond the initial number of alive cells.

The definition includes the possibility that a glider gun (or a similar pattern) is created which will result in an unbounded number of alive cells.

Links

Table of n, a(n) for n=1..62.

Wikipedia, Ulam spiral.

Wikipedia, Conway's Game of Life.

Example

n=1 to n=4 die out very quickly (within 3 steps). The maximum number of alive cells is simply the number of alive cells in the initial pattern, i.e., n.
n=5 is the first term that leads to somewhat interesting steps in the game of life simulation (although the maximum number of alive cells still does not exceed the initial number 5):
  . . . . . | . . . . . | . . . o . | . . . o . | . . . o . | . . . . .
  o . o . . | . o o o . | . . o . o | . . o . o | . . . o . | . . . . .
  . . o o . | . . o o . | . . o . o | . . . . . | . . . . . | . . . . .
  o . . . . | . . . . . | . . . . . | . . . . . | . . . . . | . . . . .
n=8 leads to a maximum number of 65 alive cells and stabilizes after 107 steps. Initial pattern:
  o . . . o |
  . o . o . |
  o . . o o |
  . o . . . |
n=15 reaches a maximum of 160 alive cells and is the first pattern that leads to having a glider (escaping in the northeast direction). Besides the glider, the stabilized pattern contains 4 blinkers, 3 blocks, 2 beehives and 1 ship.

Crossrefs

Cf. A214664, A214665.

Cf. A152389, A126803, A099733, A179412.

Sequence in context: A043312 A171635 A044909 * A273475 A278942 A303166

Adjacent sequences: A370773 A370774 A370775 * A370777 A370778 A370779

Keyword

nonn

Author

Thomas Strohmann, Mar 01 2024

Status

approved